CS70 Guide
  • This site is now deprecated
  • LaTeX Reference
  • Discrete Math
    • Overview
    • Propositional Logic
    • Proofs
    • Stable Matching
    • Graphs
    • Modular Arithmetic
    • RSA Cryptography
    • Polynomials
    • Countability
    • Computability
  • Probability
    • Overview
    • Counting
    • Discrete Probability
    • Hashing and the Union Bound
    • Expectation and Variance
    • Concentration Inequalities
    • Continuous Probability
    • Markov Chains
    • The Beta Family
    • The Gamma Family
    • Conditional Expectation and Variance
Powered by GitBook
On this page

Was this helpful?

  1. Probability

Expectation and Variance

PreviousHashing and the Union BoundNextConcentration Inequalities

Last updated 2 years ago

Was this helpful?

The expectation of a random variable, E(X)E(X)E(X), is the average of possible values weighted by their probabilities. Formally, it can be defined in two ways:

  1. Domain definition: E(X)=∑ω∈ΩX(ω)P(ω)E(X) = \sum_{\omega \in \Omega} X(\omega) P(\omega)E(X)=∑ω∈Ω​X(ω)P(ω).

  2. Range definition: E(X)=∑xxP(X=x)E(X) = \sum_x x P(X = x)E(X)=∑x​xP(X=x).

Expectation has nice properties of linearity: E(X+Y)=E(X)+E(Y)E(X + Y) = E(X) + E(Y)E(X+Y)=E(X)+E(Y) and E(aX+b)=aE(x)+bE(aX + b) = aE(x) + bE(aX+b)=aE(x)+b.

http://prob140.org/textbook/content/Chapter_08/01_Definition.html